Significant Figures Calculator

Which zeros count? The four rules for significant figures

Every nonzero digit is always significant. Zeros are the tricky part, and follow three rules: a leading zero (before the first nonzero digit, like the two zeros in 0.00405) is never significant; a captive zero (sandwiched between two significant digits, like the 0 in 405) is always significant; and a trailing zero (after the last nonzero digit) is significant only if the number has a decimal point — otherwise it's ambiguous.

For 0.004050: the two zeros before the 4 are leading zeros (not significant), the 0 between the 4 and 5 is captive (significant), and the final 0 is trailing but the number has a decimal point, so it's significant too — giving 4 significant figures in total.

Why 1,500 and 1,500. don't have the same number of significant figures

Without a decimal point, trailing zeros in a whole number are genuinely ambiguous — 1500 could mean anything from 2 to 4 significant figures depending on how precisely it was actually measured, so by convention only the nonzero digits (and any captive zeros) count: 2 significant figures.

Writing a decimal point removes the ambiguity entirely — 1500. unambiguously has 4 significant figures, because a decimal point signals that every digit shown, including the trailing zeros, was actually measured or known precisely. Scientific notation (1.500 × 10³) works the same way and is the least ambiguous option of all.

A number with both a captive and a trailing zero: 120.050

120.050 demonstrates every rule at once: reading left to right, 1 is significant (first nonzero digit), 2 is significant, the first 0 is captive because more significant digits follow later in the number, 0 is captive again, 5 is significant, and the final 0 is a trailing zero made significant by the decimal point. That's 6 significant figures — every single digit in the number.

Rounding to N significant figures without the floating-point trap

To round to N significant figures, count from the first significant digit, keep N digits, and look at the very next digit: 5 or higher rounds the last kept digit up, below 5 leaves it unchanged. Rounding 0.004050 to 2 significant figures: the first two significant digits are 4 and 0, and the next digit is 5, so the second digit rounds up — giving 0.0041.

This calculator rounds using the actual digits you typed, not a raw floating-point shortcut — because 0.004050 can't be represented exactly in binary floating point, a naive rounding approach can silently round the wrong way. Working from the digit string avoids that entirely.

The operation rules: multiplying/dividing vs. adding/subtracting

Multiplication and division: the result can't have more significant figures than the least-precise input. 12.11 has 4 sig figs and 18 has 2 — the product is rounded to the smaller count, 2 significant figures (limited by 18).

Addition and subtraction: the rule is about decimal places, not sig figs — the result can't have more decimal places than the least-precise input. Adding 12.11 + 18.0 + 1.013: the fewest decimal places among the inputs is 1 (from 18.0), so the raw sum 31.122999999999998 rounds to 31.1.